“surely Moab will be as Sodom, and the children of Ammon as Gomorrah.” The cities of Sodom and Gomorrah were not picked at random. God could have used any number of examples of destruction, such as the destruction at the time of the Flood, the destruction of Egypt, the fall of Jericho, etc. But the countries of Moab and Ammon overlooked the Dead Sea region to the west of them, and the sites where Sodom and Gomorrah had been would have been clearly visible (those sites are now lost in history). So Sodom and Gomorrah should have been a warning to Moab and Ammon not to defy God, but that warning went unheeded, and eventually Moab and Ammon were mostly destroyed, a destruction that will be completed in the Great Tribulation and Battle of Armageddon.
“and the survivors of my nation will inherit them.” This can be confusing because according to Ezekiel 47 and 48, in the Millennial Kingdom the tribes of Israel will be on the west side of the Jordan River, while traditionally Moab and Ammon were on the east side of the Jordan. But early on in history when Moab and Ammon were born as sons of Abraham’s nephew Lot, they were born on the west side of the Jordan River, in the territory that would become Israel, because Abraham conceded that territory to Lot (Gen. 13:10-11).